{"paper":{"title":"Mixed Hermitian volume and number of common zeros of holomorphic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Boris Kazarnovskii","submitted_at":"2018-11-14T11:36:13Z","abstract_excerpt":"Let $V_i$ be a finite dimensional Hermitian vector space of holomorphic sections of a line bundle $L_i$ on a complex $n$-dimensional manifold $X$. We associate to $V_i$ the non-negative Hermitian quadratic form $g_i$ on $X,$ define a Hermitian mixed volume of $X$ for a \"mixing tuple\" of $n$ non-negative Hermitian forms, and prove that the average number of common zeroes of $f_1\\in V_1,\\ldots, f_n\\in V_n$ equals to the mixed volume of $X$ for the \"mixing tuple\" $g_1,\\ldots,g_n$. This note is related to arXiv:1802.02741, where the average number of common zeros for real equations are treated in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05733","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}